Spectra, exit times and long time asymptotics in the zero-white-noise limit

Abstract

For ϵ>0, we consider the valued Markov process whose generator is . X ϵ is a small random perturbation of the deterministic dynamical system . We first establish rough asymptotics for the eigenvalues of the operator with Dirichlet of Neumann boundary conditions when ϵ tends to 0. Using these estimates, we describe the asymptotics of where is the exit time of X ϵ out of a fixed bounded domain. We are also able to express the limit law of X ϵ at time exp as a linear combination of the equilibrium measure of X ϵ restricted to some domains of . We derive a more precise description of X ϵ giving sufficient conditions for the unpredictability of and metastability. Our approach is based on spectral estimates and ultra-contractivity